Theorem fmptapd 7107

Description: Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) (Revised by AV, 10-Aug-2024.)

Hypotheses

Ref	Expression
fmptapd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fmptapd.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
fmptapd.s	⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
fmptapd.c	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵)

Assertion

Ref	Expression
fmptapd	⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑆   𝜑,𝑥

Allowed substitution hints:   𝐶(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem fmptapd

Step	Hyp	Ref	Expression
1		fmptapd.c	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵)
2		fmptapd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		fmptapd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
4	1, 2, 3	fmptsnd 7105	. . 3 ⊢ (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶))
5	4	uneq2d 4119	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
6		mptun 6628	. . 3 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
7	6	a1i 11	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
8		fmptapd.s	. . 3 ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
9	8	mpteq1d 5182	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶))
10	5, 7, 9	3eqtr2d 2770	1 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∪ cun 3901  {csn 4577  ⟨cop 4583   ↦ cmpt 5173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-opab 5155  df-mpt 5174

This theorem is referenced by:  fmptpr  7108  poimirlem3  37603  poimirlem4  37604  poimirlem16  37616  poimirlem17  37617  poimirlem19  37619  poimirlem20  37620